A61B5/00—Detecting, measuring or recording for diagnostic purposes; Identification of persons

A61B5/04—Measuring bioelectric signals of the body or parts thereof

A61B5/0402—Electrocardiography, i.e. ECG

A61B5/0408—Electrodes specially adapted therefor

A61B5/04085—Multiple electrode holders

Abstract

A method of diagnosing an abnormal condition in a biological structure, such
as the heart, including the steps of measuring a physiological response at at least three
sampled points on a surface of the biological structure, calculating a vector function
related to the response, displaying a representation of the vector function, and
inferring the abnormal condition from the representation. The present invention is
particularly useful for diagnosing cardiac arrhythmias, in which case the physiological
response is a voltage, from which is inferred a local activation time and the vector
function is a gradient of the local activation time, specifically, a conduction velocity.

Description

FIELD AND BACKGROUND OF THE INVENTION

The present invention relates generally to systems and methods for mapping,
and specifically to methods of mapping of intrabody organs.

Cardiac mapping is used to locate aberrant electrical pathways and currents
within the heart, as well as mechanical and other aspects of cardiac activity. Various
methods and devices have been described for mapping the heart. Such methods and
device are described, for example, in U.S. patents 5,471,982, 5,391,199 and 5,718,241
and in PCT patent publications WO94/06349, WO96/05768 and WO97/24981. U.S.
patent 5,391,199, for example, describes a catheter including both electrodes for
sensing cardiac electrical activity and miniature coils for determining the position of
the catheter relative to an externally-applied magnetic field. Using this catheter a
cardiologist may collect a set of sampled points within a short period of time, by
determining the electrical activity at a plurality of locations and determining the
spatial coordinates of the locations.

In order to allow the surgeon to appreciate the determined data, a map,
preferably a three dimensional (3D) map, including the sampled points is produced.
U.S. patent 5,391,199 suggests superimposing the map on an image of the heart. The
positions of the locations are determined with respect to a frame of reference of the
image. However, it is not always desirable to acquire an image, nor is it generally
possible to acquire an image in which the positions of the locations can be found with
sufficient accuracy.

Various methods are known in the art for reconstructing a 3D map of a cavity
or volume using the known position coordinates of a plurality of locations on the
surface of the cavity or volume. Some methods include triangulation, in which the
map is formed of a plurality of triangles which connect the sampled points. In some
cases a convex hull or an alpha-hull of the points is constructed to form the mesh, and
thereafter the constructed mesh is shrunk down to fit on the sampled points within the
hull. Triangulation methods do not provide a smooth surface and therefore require
additional stages of smoothing.

Another method which has been suggested is forming a bounding ellipsoid
which encloses the sampled points. The sampled points are projected onto the
ellipsoid, and the projected points are connected by a triangulation method. The
triangles are thereafter moved with the sampled points back to their original locations,
forming a crude piecewise linear approximation of the sampled surface. However, this
method may reconstruct only surfaces which have a star shape, i.e., a straight line
connecting a center of the reconstructed mesh to any point on the surface does not
intersect the surface. In most cases heart chambers do not have a star shape.

In addition, reconstruction methods known in the art require a relatively large
number of sampled locations to achieve a suitable reconstructed map. These methods
were developed, for example, to work with CT and MRI imaging systems which
provide large numbers of points, and therefore generally work properly only on large
numbers of points. In contrast, determining the data at the locations using an invasive
catheter is a time-consuming process which should be kept as short as possible,
especially when dealing with a human heart. Therefore, reconstruction methods which
require a large number of determined locations are not suitable.

One important example of cardiac mapping is the determination of the speed
and direction of propagation of electrical signals through the tissue of the heart.
Abnormal propagation velocity, or vortical signal flow, may be diagnostic of locally
diseased heart tissue that should be treated, for example by ablation. Typically, the
velocity of propagation of cardiac signals is measured by sensing the wavefronts at a
plurality of electrodes in contact with the inner surface of a chamber of the heart. A
representative example of the prior art in this field is Kadish, et al., "Vector Mapping
of Myocardial Activation", Circulation, Vol. 74, No. 3, Pages 603-615 (September
1986), in which vectors based on activation maps are drawn perpendicular to the
isochrome tangent. Kadish et al. describes the measurement of the timing of local
depolarization events, using an array of electrodes, for the purpose of deriving
propagation velocities. This propagation velocity deriving technique is also described
in Gerstenfeld et al., "Evidence for Transient Linking of Atrial Excitation During
Atrial Fibrillation in Humans", Circulation, Vol. 86, No. 2, Pages 375-382 (August
1992) and Gerstenfeld et al., "Detection of Changes in Atrial Endocardial Activation
with Use of an Orthogonal Catheter", J. Am. Coll. Cardiol. 1991; 18:1034-42 as well
as U.S. Patent 5,487,391 (Panescu).

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an improved method for
mapping a 3D volume or cavity, based on the positions of points on a surface of the
volume or cavity.

It is an object of some aspects of the present invention to provide methods and
apparatus for generating a map of a volume in the human body from a plurality of
sampled points, regardless of the shape of the volume.

It is another object of some aspects of the present invention to provide a
simple, rapid method for reconstructing a 3D map of a volume in the human body
from a plurality of sampled points, preferably using fewer sampled points than is
feasible using methods known the art.

It is another object of preferred embodiments of the present invention to
provide a method for reconstructing a 3D map of a volume in the human body from a
plurality of sampled points, without assuming any topological relationship between
the points.

It is another object of some aspects of the present invention to provide a
simple method for reconstructing a 3D map of a volume in movement.

It is another object of some aspects of the present invention to provide a
simple method for reconstructing a 3D map of a volume in the human body from a
plurality of sampled points independent of the sampling order.

It is another object of some aspects of the present invention to provide a quick
method for reconstructing a 3D map of a volume in the human body from a plurality
of sampled points, such that the method may be used in interactive procedures.

It is another object of some aspects of the present invention to provide a
method for reconstructing a smooth 3D map of a volume in the human body from a
plurality of sampled points.

In preferred embodiments of the present invention, a processor reconstructs a
3D map of a volume or cavity in a patient's body (hereinafter referred to as the
volume), from a plurality of sampled points on the volume whose position coordinates
have been determined. In contrast to prior art reconstruction methods in which a large
number of sampled points are used. the preferred embodiments of the present
invention are directed to reconstruction of a surface based on a limited number of
sampled points. The number of sampled points is generally less than 200 points and
may be less than 50 points. Preferably, ten to twenty sampled points are sufficient in
order to perform a preliminary reconstruction of the surface to a satisfactory quality.

An initial, generally arbitrary, closed 3D curved surface (also referred to
herein for brevity as a curve) is defined in a reconstruction space in the volume of the
sampled points. The closed curve is roughly adjusted to a shape which resembles a
reconstruction of the sampled points. Thereafter, a flexible matching stage is
preferably repeatedly performed once or more to bring the closed curve to accurately
resemble the shape of the actual volume being reconstructed. Preferably, the 3D
surface is rendered to a video display or other screen for viewing by a physician or
other user of the map.

In preferred embodiments of the present invention, the initial closed curved
surface encompasses substantially all the sampled points or is interior to substantially
all the sampled points. However, it is noted that any curve in the vicinity of the
sampled points is suitable. Preferably, the closed 3D curved surface comprises an
ellipsoid, or any other simple closed curve. Alternatively, a non-closed curve may be
used, for example, when it is desired to reconstruct a single wall rather than the entire
volume.

A grid of a desired density is defined on the curve, and adjustment of the curve
is performed by adjusting the grid points. The grid preferably divides the curved
surface into quadrilaterals or any other polygons such that the grid evenly defines
points on the curve. Preferably, the grid density is sufficient such that there are
generally more grid points than sampled points in any arbitrary vicinity. Further
preferably, the grid density is adjustable according to a desired compromise between
reconstruction accuracy and speed.

In some preferred embodiments of the present invention, external information
is used to choose an initial closed curve which is more closely related to the
reconstructed volume, for example, using the image of the volume, as described
above. Thus, the reconstruction procedure may produce a more accurate
reconstruction in less time. Alternatively or additionally, a database of closed curves
suitable for various volumes of the body is stored in a memory, and the curve to be
used is chosen according to the specific procedure. In a further preferred embodiment
of the present invention, a map of a reconstructed volume in a patient is used as a
beginning curve for subsequent mapping procedures performed at later times on the
same volume.

Preferably, the rough adjustment of the closed curve is performed in a single
iteration, most preferably by calculating for each grid point an adjustment point, and
moving the grid point a fraction of the distance to the adjustment point. Preferably,
the grid point is moved about 50-80% of the distance between its original point and
the adjustment point, more preferably about 75%.

The adjustment point is preferably determined by taking a weighted sum over
substantially all the sampled points. Preferably, the weights are inversely related to the
distances from the adjusted grid point to the sampled points, referred to herein as grid
distances. In a preferred embodiment of the present invention, each weight is defined
as the reciprocal of the sum of a small constant plus the grid distance, raised to a
predetermined power, so that sampled points close to the grid point are given a larger
weight. Preferably, the power is approximately between 4 to 9, most preferably 8. The
small constant is preferably smaller than the magnitude of the smallest grid distance,
and is preferably of the size of the accuracy of the determination of the coordinates of
the sampled points. The small constant is used to prevent division by zero when a
grid-point is on a sampled point.

In some preferred embodiments of the present invention, the weights also
include a factor which is indicative of the density of points in the vicinity of their
corresponding point. Preferably, the weight is multiplied by a density value between
zero and one, indicative of the density, such that isolated sampled points influence the
sum more than sampled points in a dense area. Preferably, the influence of the points
is thus substantially independent of the density of points in their vicinity.

In a preferred embodiment of the present invention, the flexible matching step
is performed by associating each sampled point with a corresponding grid-point, such
that each sampled point is associated with the grid point which is closest to it. A
movement vector is calculated for each of the associated and non-associated grid-points.
Preferably, the movement vectors are calculated based on vectors from the
associated grid points to their respective sampled points. Further preferably, the
sampled points influence the value of the movement vector for a specific point
according to their proximity to the specific point. In addition, the function by which
the movement vectors are calculated is preferably smooth and does not include
complicated calculations. Preferably, the function is a weighted sum of the vectors
from the associated grid points to their respective sampled points. The grid points are
then moved according to their respective movement vectors.

Additionally or alternatively, the associated grid points are moved toward their
corresponding sampled points by a percentage of the distance between them. Those
grid points which are not associated with a sampled point are moved a distance which
is determined by interpolation between the distances which surrounding points on the
grid are moved. Preferably, the resulting grid is smoothed using a suitable smoothing
transformation. Preferably, the process of associating and moving is repeated two or
more times to allow finer adjustment of the closed curve.

In a preferred embodiment of the present invention, a user can adjust the
number of times the flexible matching step is repeated according to a desired
compromise between image quality and speed. Alternatively or additionally, a quick
reconstruction is first provided to the user, and thereafter the calculation is repeated to
receive a finer reconstruction. Preferably, the weights of the weighted sum used in the
flexible matching stage are adjusted according to the number of times the matching is
to be performed. Alternatively or additionally, the weights are determined for each
flexible matching step according to its place in the sequential order of the flexible
matching steps.

Preferably, the distances used for the weights and/or for interpolation are
Euclidean geometrical distances between the points. The Euclidean distance is easily
computed and causes points on opposite walls of the volume to mutually repel, so that
the walls do not intersect. Alternatively, other distances, such as the distance along the
original or adjusted grid, may be used. In a preferred embodiment of the present
invention, during the first flexible matching step the distance used is the distance
along the original grid while subsequent flexible matching steps use the Euclidean
distance.

In some preferred embodiments of the present invention, a smoothing process
is applied to the reconstructed surface, preferably by applying a surface convolution
with a Gaussian-like kernel. The smoothing process provides a better approximation
of the surface and allows easier performance of calculations based on the
reconstructed surface. However, applying the surface convolution results in some
shrinkage of the surface, and therefore an affine transformation is preferably
performed on the smoothed surface. The affine transformation is preferably chosen
according to those sampled points which are external to the reconstructed surface. The
chosen affine transformation preferably minimizes the mean square distance of the
external points to the surface.

Preferably, when the reconstruction is finished, each sampled point
substantially coincides with a grid point. In some preferred embodiments of the
present invention, a final exact matching stage is performed. Each sampled point is
associated with a closest grid point, and the associated grid point is moved onto the
sampled point. The rest of the grid points are preferably not moved. Generally, most
of the sampled points are by this stage very close to the reconstructed surface, and
therefore the smoothness of the surface is substantially not affected. However, some
outlier sampled points, i.e., sampled points which do not belong to the surface, may
cause substantial changes to the surface. Preferably, the user may determine whether
to move the surface onto points that are distanced from the surface by more than a
predetermined maximum distance. Alternatively or additionally, the entire exact
matching step is optional and is applied only according to a user request.

Further alternatively or additionally, the grid points are brought to a fixed
distance from the sampled points. Leaving such a fixed distance may be desired, for
example, when the sampled coordinates are of locations close to a distal tip of a
sampling catheter rather than at the distal tip itself.

In preferred embodiments of the present invention, data regarding the sampled
points are acquired by positioning a catheter within the volume which is to be
reconstructed, for example, within a chamber of the heart. The catheter is positioned
with a distal end thereof in contact with each of the sampled points in turn, and the
coordinates of the points and, optionally, values of one or more physiological
parameters are sensed at a distal end of the catheter. Preferably, the catheter comprises
a coordinate sensor close to its distal end, which outputs signals indicative of the
coordinates of the tip of the catheter. Preferably, the coordinate sensor determines the
position by transmitting and receiving electromagnetic waves, as described, for
example, in PCT publications GB93/01736, WO94/04938, WO97/24983 and
WO96/05768, or in U.S. Patent 5,391,199, which are all incorporated herein by
reference.

In some preferred embodiments of the present invention, the reconstructed
volume is in movement, for example, due to beating of the heart. In such
embodiments, the sampled points are preferably registered with a reference frame
fixed to the heart. Preferably, a reference catheter is fixed in the heart, and the
sampled points are determined together with the position of the reference catheter
which is used to register the points, as described, for example, in the above-mentioned
U.S. Patent 5,391,199 and PCT publication WO96/05768.

Alternatively or additionally, when at least part of the movement is a cyclic
movement, as in the heart, acquisition of the sampled points is synchronized to a
specific time point of the cycle. Preferably, when the sampled volume is in the heart,
an ECG signal is received and is used to synchronize the acquisition of the sampled
points. For example, the sampled points may be acquired at end diastole. Further
alternatively or additionally, the coordinates of each of the sampled points are
determined together with an indication of the time point relative to the cyclic
movement in which the coordinates were acquired. Preferably, the indication includes
the relative time from the beginning of the cycle and the frequency of the cyclic
movement. According to the frequency and the relative time, the determined
coordinates are corrected to end diastole, or any other point in the cyclic movement.

In some preferred embodiments of the present invention, for each sampled
point a plurality of coordinates are determined at different time points of the cyclic
movement. In one of these preferred embodiments, each sampled point has two
coordinates which define the range of movement of the point. Preferably, if the
plurality of coordinates of different points are associated with different cycle
frequencies, the coordinates are transformed so as to correspond to a set of
coordinates in a single-frequency cyclic movement. Further preferably, the
coordinates are processed so as to reduce or substantially eliminate any contribution
due to movement other than the specific (cardiac) cyclic movement, such as
movement of the chest due to respiration. Reconstruction is performed for a plurality
of configurations of the volume at different time points of the cyclic movement.
Preferably, a first reconstruction is performed as described above to form an anchor
reconstruction surface, and reconstruction of surfaces for other time points of the
cycle are performed relative to the anchor reconstruction surface.

Preferably, for each further time point of the cyclic movement, the anchor
surface is adjusted according to the coordinates of the sampled points at the further
time point relative to the coordinates of the sampled points of the anchor surface.
Preferably, the anchor surface is adjusted by a quadratic transformation which
minimizes a mean square error, the error representing the distances between the
sampled points of the further time point and the adjusted surface. Alternatively or
additionally, an affine transformation is used instead of the quadratic transformation.
Further alternatively or additionally, a simple transformation is used for surfaces
having relatively few sampled points, while surfaces with a relatively large number of
sampled points a quadratic transformation is used. The simple transformation may be
an affine transformation, a scaling and rotation transformation, a rotation
transformation, or any other suitable transformation.

Preferably, the adjustment of the surface for the further time points includes,
after the transformation, one or more, preferably two, flexible matching steps and/or
an exact matching stage.

Alternatively or additionally, the reconstruction is performed separately for
each of the further time points. Further alternatively or additionally, a first
reconstruction of the surfaces for the further time points is performed relative to the
anchor surface, and afterwards a more accurate reconstruction is performed for each
time point independently.

In some preferred embodiments of the present invention, dedicated graphics
hardware which is designed to manipulate polygons is used to perform the
reconstruction stages described above.

In some preferred embodiments of the present invention, one or more
physiological parameters are acquired at each sampled point. The physiological
parameters for the heart may comprise a measure of cardiac electrical activity, for
example, and/or may comprise any other type of local information relating to the
heart, as described in the above-mentioned PCT patent publication WO97/24981,
which is incorporated herein by reference. The one or more physiological parameters
may be either scalars or vectors and may comprise, for example, a voltage,
temperature, pressure, impedance, conduction velocity, or any other desired value.

It is noted that the physiological response is a time of arrival of a physiological
signal propagating in the biological structure and the vector function may be any of a
number of vector functions (as noted above). For example, the vector function may
be a conduction velocity of the electrical activity.

Preferably, after the volume is reconstructed based on the coordinates, values
of the physiological parameter are determined for each of the grid points based on
interpolation of the parameter value at surrounding sampled points. Preferably, the
interpolation of the physiological parameter is performed in a manner proportional to
the aggregate interpolation of the coordinates. Alternatively, the physiological
parameters are interpolated according to the geometrical distance between the points
on the grid. Alternatively or additionally, the physiological parameters are
interpolated in a manner similar to the flexible matching step described hereinabove.

The reconstructed surface may be displayed in movement, and/or a physician
may request a display of a specific time point of the cycle. Preferably, the
physiological parameter is displayed on the reconstructed surface based on a
predefined color scale. In a preferred embodiment of the present invention, the
reliability of reconstruction of regions of the reconstructed surface is indicated on the
displayed surface. Preferably, regions which are beneath a user-defined threshold are
displayed as semi-transparent, using α-blending. Preferably, the reliability at any grid
point is determined according to its proximity to sampled points. Those points on the
grid which are beyond a predetermined distance from the nearest sampled point are
less reliable.

In some preferred embodiments of the present invention, acquired images such
as LV-grams and fluoroscopic images are used together with the sampled points to
enhance the speed and/or accuracy of the reconstruction. Preferably, the processor
performs an object recognition procedure on the image to determine the shape of the
closed 3D curved surface to use in constructing the initial grid of the reconstruction.
Alternatively or additionally, the image is used by the physician to select areas in
which it is most desired to receive sampled points.

In some preferred embodiments of the present invention, the physician may
define points, lines, or areas on the grid which must remain fixed and are not to be
adjusted. Alternatively or additionally, some points may be acquired as interior points
which are not to be on the map since they are not on a surface of the volume. The
reconstruction procedure is performed accordingly so that the closed curve is not
moved too close to the interior points.

In some preferred embodiments of the present invention, the reconstruction
surface is used to determine an accurate estimate of the volume of the cavity. The
surface is divided by the grid points into quadrilaterals, and each quadrilateral is
further divided into two triangles. Based on these triangles the volume defined by the
surface is estimated. Alternatively, the volume is calculated using a volumetric
representation. Other measurements, such as geodesic surface measurements on the
surface, may also be performed using the reconstructed surface.

It is noted that some of the stages described above may be ignored in some
preferred embodiments of the invention, in order to save processing time and speed up
the reconstruction procedure.

One example of a physiological parameter to which the present invention is
particularly applicable is the local activation time (LAT) of heart tissue. The present
invention allows the measurement of LAT, relative to the cardiac cycle, at a plurality
of sampled points on the inner surface of a chamber of the heart, using a device, at the
tip of a catheter, that senses electrical activity only at a single point of contact of the
catheter tip with the inner surface of the chamber of the heart. These measurements of
LAT are posted at corresponding points on a grid that corresponds to a particular time
in the cardiac cycle, preferably end diastole, and are interpolated to the other grid
points. The grid points define polygons, for example, triangles; and a vectorial
propagation velocity is determined for each grid polygon from the LAT values at the
grid points that are the vertices of the polygon. Each grid then is assigned the average
of the propagation velocities of the polygon of which it is a vertex, and the
propagation velocities at the grids are smoothed and displayed, preferably as arrows
posted at the grid points, with the directions of the arrows representing the direction of
propagation and the lengths of the arrows representing the speed of propagation.
These arrows provide a visual display of propagation speed and propagation vorticity
that enables an electrophysiologist to identify the location of diseased cardiac tissue
that should be treated. Note that this measurement and display of propagation
velocity is based on consecutive measurements at individual points on the inner
surface of the chamber of the heart, unlike the prior art methods, which require
simultaneous measurements at at least two distinctly separated points.

More generally, such a display may be constructed for any vector function that
is related to a physiological response measured at discrete points on the surface of a
biological structure. The vector function may be any of a number of vector functions.
For example, the vector function may be a conduction velocity of the physiological
response.

LAT is the time interval between a reference time determined, for example,
from the body surface ECG or intracardiac electrogram, and the time of the local
depolarization event. Other useful scalar functions of the physiological parameters,
may be calculated and displayed, superposed on a combined display of LAT (as
pseudocolor) and propagation velocity (as arrows). One such useful scalar function is
the range of voltages measured at each sampled point (displayed as a pseudocolor): an
abnormally low range is diagnostic of scar tissue, upon which the conduction velocity
may be displayed as arrows.

There is therefore provided in accordance with a preferred embodiment of the
present invention, a method of reconstructing a map of a volume, including
determining coordinates of a plurality of locations on a surface of the volume having a
configuration, generating a grid of points defining a reconstruction surface in 3D
space in proximity to the determined locations, for each of the points on the grid,
defining a respective vector, dependent on a displacement between one or more of the
points on the grid and one or more of the locations, and adjusting the reconstruction
surface by moving substantially each of the points on the grid responsive to the
respective vector, so that the reconstruction surface is deformed to resemble the
configuration of the surface.

Preferably, the method includes displaying the reconstruction surface.

Preferably, generating the grid includes generating a grid such that the
reconstruction surface encompasses substantially all of the determined locations or is
interior to substantially all of the determined locations.

Preferably, generating the grid includes defining an ellipsoid.

Preferably, the reconstruction surface is defined and adjusted substantially
independently of any assumption regarding a topology of the volume.

Further preferably, the reconstruction surface is defined and adjusted
substantially without reference to any point within the volume.

Alternatively or additionally, generating the grid includes acquiring an image
of the volume and defining the reconstruction surface such that it resembles the image
of the volume.

Further alternatively or additionally, generating the grid includes choosing a
grid from a memory library according to at least one characteristic of the volume.

Preferably, adjusting the surface includes a rough adjustment stage and a
flexible matching stage.

Preferably, the rough adjustment stage includes moving each point on the grid
toward a respective weighted center of mass of the determined locations, and locations
closer to the point on the grid are given larger weight.

Preferably, moving each point in the rough adjustment stage includes defining,
for each of the points on the grid, a respective rough adjustment vector which includes
a weighted sum of vectors from the point to each of the determined locations and
moving the points a distance proportional to the respective vector.

Preferably, defining the rough adjustment vector includes calculating a weight
for each of the summed vectors that is generally inversely proportional to a magnitude
of the summed vector raised to a predetermined power.

Preferably, the weight includes an inverse of a sum of a constant and the
magnitude of the vector raised to a power between 4 and 10.

Preferably, the constant is smaller than a precision of the location
determination.

Preferably, moving each point includes moving each point toward a respective
target point by a distance between 50 and 90% of the distance between the point and
the target point.

Preferably, the flexible matching stage includes selecting a grid point to be
associated respectively with each of the determined locations.

Preferably, selecting the grid point includes finding for each determined
location a point on the grid that is substantially closest thereto.

Preferably, the flexible matching stage includes moving grid points that were
not selected by an amount dependent on the movements of surrounding grid points.

Preferably, moving the grid points that were not selected includes moving the
grid points by an amount dependent substantially only on the movements of
surrounding selected grid points.

Preferably, moving the grid points includes calculating a movement of a grid
point that was not selected based on the movements of the surrounding selected grid
points and distances from these surrounding grid points.

Preferably, calculating the movement of the grid point includes interpolating
between the movements of surrounding grid points.

Preferably, the distances include geometrical distances.

Alternatively or additionally, the distances include a length of the
reconstruction surface between the grid points.

Preferably, the flexible matching stage includes defining a displacement
function which includes a weighted sum of vectors, each vector connecting a location
and its associated point.

Preferably, the flexible matching stage includes moving the grid points
according to the displacement function so as to smooth the surface.

Preferably, determining the coordinates includes positioning a catheter tip at
the plurality of locations.

Preferably, positioning the catheter tip includes positioning the catheter at a
plurality of locations in a chamber of the heart.

Preferably, determining the coordinates includes positioning a catheter tip at
the plurality of locations.

Preferably, determining the coordinates includes transmitting and receiving
non-ionizing waves.

Preferably, determining the coordinates includes positioning at the plurality of
locations a device which generates signals indicative of the position of the device.

Preferably, the device generates signals indicative of the six degrees of
position and orientation of the device.

Preferably, determining the coordinates includes receiving the coordinates
from an external source.

Preferably, the method includes acquiring a signal indicative of a value of
physiological activity at substantially each of the plurality of locations.

Preferably, acquiring the signal includes acquiring a signal indicative of a
value of electrical activity at the location.

Preferably, the method includes estimating a value of the physiological
activity at the adjusted grid points.

Preferably, estimating the value of the physiological activity includes
estimating based on an acquired value of the physiological activity at a location in a
vicinity of the adjusted grid points.

Preferably, estimating based on the acquired value includes interpolating the
value responsive to deformation of the reconstruction surface.

Preferably, determining coordinates of a plurality of locations includes
determining coordinates of less than 200 locations, more preferably of less than 50
locations, and most preferably of less than 20 locations.

Preferably, the volume is in motion, and determining the coordinates includes
determining a correction factor responsive to the motion.

Preferably, the motion includes cyclic motion, and determining the correction
factor includes determining a factor responsive to a cycle frequency of the motion.

Preferably, determining the factor includes filtering out motion at a frequency
substantially different from the cycle frequency.

Preferably, the motion includes cyclic motion, and determining the coordinates
includes determining the coordinates at a predetermined phase of the cyclic motion.

Preferably, determining the coordinates at the predetermined phase includes
determining the coordinates in a plurality of time points and adjusting the coordinates
relative to the cyclic movement.

Preferably, adjusting the coordinates includes determining a rate of the cyclic
movement together with the coordinates for substantially each coordinate
determination.

Preferably, generating the grid and adjusting the reconstruction surface are
performed separately with respect to the coordinates determined in each phase of the
cyclic motion.

Alternatively or additionally, generating and adjusting are performed for the
coordinates of a plurality of phases of the cyclic motion so as to form a motion map of
the volume.

Preferably, generating the grid and adjusting the reconstruction surface are
performed for a first group of coordinates determined in a first phase of the cyclic
motion, and the reconstructed surface of the first group is adjusted to form a
reconstructed surface in one or more additional phases.

Preferably, the method includes smoothing the reconstructed surface.

Preferably, the method includes applying an affine transformation to the
reconstructed surface.

Preferably, the method includes a final stage in which each determined
location is associated with a respective grid point, and the associated grid points are
moved onto the determined locations while non-associated grid points are
substantially not moved.

Preferably, the method includes estimating a measure of the volume
responsive to the reconstructed surface.

Preferably, estimating the measure of the volume includes choosing an
arbitrary point inside the grid and calculating the volumes of tetrahedrons defined by
the arbitrary point and groups of three points on the grid which cover the entire grid
surface.

There is further provided in accordance with a preferred embodiment of the
present invention, apparatus for reconstructing a map of a volume from coordinates of
a plurality of determined locations on a surface of the volume having a configuration,
including a processor, which receives the coordinates and generates a grid of points
defining a reconstruction surface in 3D space in proximity to the determined
locations, and which defines a respective vector for each of the points on the grid,
dependent on a displacement between one or more of the points on the grid and one or
more of the locations, and which adjusts the reconstruction surface by moving each of
the points on the grid responsive to the respective vector, so that the reconstruction
surface is deformed to resemble the configuration of the surface of the volume.

Preferably, the apparatus includes a display screen for displaying the adjusted
surface.

Preferably, the processor analyzes the adjusted surface to determine a
characteristic of the volume.

Preferably, the apparatus includes a memory for storing the adjusted surface.

Preferably, the grid initially encompasses substantially all of the determined
locations.

Preferably, the apparatus includes an imaging device for acquiring an image of
the volume, and the processor defines the grid initially such that it resembles the
image of the volume.

Preferably, the apparatus includes a memory library including a plurality of
closed curves, and the processor defines the grid initially by choosing a closed curve
from the memory library according to at least one characteristic of the volume.

Preferably, the processor generates and defines the reconstruction surface
substantially independently of any assumption regarding a topology of the volume.

Preferably, the processor generates and defines the reconstruction surface
substantially without reference to any point within the volume.

Preferably, the processor forms the adjusted grid in two stages: a rough
adjustment stage and a flexible matching stage.

Preferably, in the rough adjustment stage, the processor moves each point on
the grid toward a respective weighted center of mass of the determined locations, and
locations closer to the point on the grid are given larger weight.

Preferably, the processor calculates the center of mass using a weight that is
substantially proportional for each location to the inverse of the sum of a small
constant and the distance between the point and the location raised to a power
between 4 and 10.

Preferably, the constant is smaller than a precision of the location
determination.

Preferably, in the flexible matching stage, the processor selects a respective
grid point to associate with each of the determined locations.

Preferably, the selected grid point for each determined location includes a
point on the grid that is closest to the location.

Preferably, the processor moves non-selected grid points by an amount
dependent on the movements of surrounding grid points.

Preferably, the amount of movement of the non-selected grid points is
dependent on the movements of surrounding selected grid points.

Preferably, the amount of movement of each of non-selected grid points is
calculated by the processor based on the distances from the surrounding selected grid
points to the non-selected grid point.

Preferably, the amount of movement of the non-associated grid points is
calculated by the processor based on an interpolation of the movements of
surrounding selected grid points.

Preferably, the distances include geometrical distances.

Preferably, the apparatus includes a probe, which is brought into engagement
with the surface to determine the locations thereon.

Further preferably, the probe includes a position sensor which indicates the
position of a tip of the probe.

Preferably, the sensor includes at least one coil.

Preferably, the sensor generates signals indicative of position and orientation
of the sensor.

Alternatively or additionally, the probe includes a functional portion for
acquiring a value of a physiological activity at the plurality of locations.

Preferably, the functional portion includes an electrode.

Preferably, the processor estimates a value of the physiological activity at the
adjusted grid points.

Preferably, the processor estimates the value of the physiological activity
based on the acquired values of the physiological activity at points surrounding the
adjusted grid points.

Preferably, the processor estimates the value by interpolation from the
acquired values responsive to deformation of the reconstruction surface.

Preferably, the apparatus includes a reference catheter for registering the
determined locations relative to a frame of reference associated with the volume.

Preferably, the apparatus includes an ECG monitor for gating the operation of
the probe so as to determine the points at a fixed phase of a cyclic movement of the
volume.

There is further provided in accordance with a preferred embodiment of the
present invention, a method of displaying values of a parameter which varies over a
surface, including determining a value of the parameter at each of a plurality of points
on the surface, and rendering an image of the surface to a display with a different
degree of transparency in different areas of the surface, responsive in each of the areas
to the value of the parameter at one or more points in the area.

Preferably, determining the value includes sampling a plurality of points and
creating a map of the surface responsive thereto, and rendering the image includes
rendering a graphic representation of the map.

Preferably, creating the map includes creating a three-dimensional map.

Preferably, determining the value includes determining a measure of reliability
of the map in each of the areas.

Preferably, rendering the image includes rending one or more of the areas
having a low measure of reliability relative to another one or more of the areas with a
relatively greater degree of transparency.

Preferably, determining the measure of reliability includes determining a
density of the sampled points.

Preferably, rendering the image includes defining a color scale and displaying
a color associated with the value, at each of the plurality of points.

Preferably, the plurality of points includes sampled points and interpolated
points, and determining the measure of reliability includes assigning a high reliability
measure to the sampled points.

Preferably, determining the measure of reliability includes assigning measures
of reliability to the interpolated points according to their respective distance from a
closest sampled point.

There is further provided in accordance with a preferred embodiment of the
present invention, a method of diagnosing a condition in a biological structure,
including measuring a physiological response at at least three sampled points on a
surface of the biological structure, calculating a vector function related to the
response, and displaying a representation of the vector function.

Preferably, the vector function is related to a gradient of the physiological
response.

Preferably, the physiological response is a function of time.

More preferably, the physiological response is a time of arrival of a
physiological signal propagating in the biological structure, and the vector function
may be any of a number of vector functions, most preferably it is a conduction
velocity.

Preferably, the representation includes an arrow at each sampled point, the
length of the arrow being related to the magnitude of the vector function at each
sampled point, and the direction of the arrow being related to the direction of the
vector function at each sampled point.

Preferably the method further includes fitting a surface to the sampled points
and displaying the surface, the display of the representation being superposed on the
display of the surface. Here, too, it is preferred that the representation includes an
arrow at each sampled point, the length of the arrow being related to the magnitude of
the vector function at each sampled point, and the direction of the arrow being related
to the direction of the vector function at each sampled point.

Preferably, the fitting of the surface to the sampled points includes
representing the surface as a grid that includes at least as many grid points as there are
sampled points. More preferably, at least one of the grid points coincides with one of
the sampled points.

Preferably, the grid includes a plurality of polygons, with the grid points being
the vertices of the polygons, each grid point being a vertex of at least one polygon,
and the calculating of the vector function includes the steps of interpolating the
response at the grid points, assigning a value of the vector function to each polygon,
with the value of the vector function assigned each polygon being based on the
interpolated response at the grid points that are the vertices of that polygon, and
determining a value of the vector function at each grid point, with the value of the
vector function at each grid point being based on the values of the vector function that
are assigned to the polygons of which that grid point is a vertex. Most preferably, the
polygons are triangles.

More preferably, calculating the vector function further includes smoothing
the values of the vector function at the grid points. Most preferably, the smoothing
parameters may be determined based on a priori knowledge about the specific heart.

Preferably, the method further includes calculating scalar functions related to
the physiological response and displaying representations of these scalar functions
superposed on the display of the surface along with the representation of the vector
function. An important example of one such scalar function is a range of the
physiological response measurements at the sampled points. In another important
example, useful in the diagnosis of heart disease, the measurements are voltage
measurements, a scalar function is the range of voltage measurements at each sampled
point, and the vector function is a conduction velocity inferred from the local
activation time.

Preferably, the physiological response is an impedance, wherein the scalar
function is a range of the impedances, and the vector function is a conduction
velocity.

Preferably, the method further includes inferring the condition from the
representation of the vector function. Preferably, inferring the condition includes
identifying at least one location on the surface that is afflicted by the condition, and
the method further includes the step of treating those locations.

Preferably, the treatment includes ablation of the biological structure at those
locations.

Preferably, the physiological response is measured consecutively at the
sampled points.

There is further provided, in accordance with a preferred embodiment of the
present invention, a method of diagnosing a condition in a biological structure,
including measuring a physiological response at at least three sampled points on a
surface of the biological structure, calculating a vector function related to the
response, and inferring the condition from the vector function.

Preferably, the vector function is related to a gradient of the physiological
response.

Preferably, the physiological response is a function of time.

More preferably, the physiological response is a time of arrival of a
physiological signal propagating in the biological structure, and the vector function is
a velocity of the propagation.

Preferably, inferring the condition includes identifying at least one location on
the surface that is afflicted by the condition, and the method further includes the step
of treating those locations.

Preferably, the treatment includes ablation of the biological structure at those
locations.

Preferably, the physiological response is measured consecutively at the
sampled points.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is herein described, by way of example only, with reference to
the accompanying drawings, wherein:

FIG. 1 is a schematic, perspective view of a heart mapping system, in
accordance with a preferred embodiment of the present invention;

FIG. 2 shows a mapping catheter within a heart of a patient, in accordance
with a preferred embodiment of the present invention;

FIG. 3 is a flow chart illustrating a method of point sampling and map
reconstruction, in accordance with a preferred embodiment of the present invention;

FIG. 4 is a flow chart illustrating a reconstruction procedure, in accordance
with a preferred embodiment of the present invention;

FIGs. 5A - 5E are simplified, two dimensional graphs illustrating
reconstruction of a map from sampled points, in accordance with a preferred
embodiment of the present invention;

FIG. 6 is a schematic illustration of a displayed reconstructed heart volume, in
accordance with a preferred embodiment of the present invention;

FIG. 7 is an illustration of a volume estimation method, in accordance with
another preferred embodiment of the present invention;

FIG. 8 is an illustration of a reconstruction procedure, in accordance with
another preferred embodiment of the present invention;

FIG. 12 shows a pattern, on a combined voltage range - conduction velocity
plot, that is diagnostic of ventricular tachycardia for a human ventricle;

FIG. 13 shows a conduction velocity display in the left ventricle of a dog
wherein the heart is entrained in a sinus rhythm from the right ventricle apex; and

FIG. 14 shows a conduction velocity display of the right atrium of a human
heart suffering from atrial flutter.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Fig. 1 shows a mapping system 18 for mapping of a volume in a patient's
body, in accordance with a preferred embodiment of the present invention. System 18
comprises an elongate probe, preferably a catheter 20, for insertion into the human
body. A distal end 22 of catheter 20 includes a functional portion 24 for performing
diagnostic and/or therapeutic functions, adjacent to a distal tip 26. Functional portion
24 preferably comprises electrodes (not shown in the figure) for performing
electrophysiological measurements, as described, for example, in U.S. patent
5,391,199 or in PCT publication WO97/24983, which are incorporated herein by
reference. Alternatively or additionally, functional portion 24 may include other
diagnostic apparatus for recording parameter values at points within the body. Such
apparatus may include a chemical sensor, a temperature sensor, a pressure sensor
and/or any other desired sensor. Functional portion 24 may determine for each point a
single value of the parameter, or alternatively a plurality of values dependent on the
time of their acquisition. Functional portion 24 may also include therapeutic
apparatus, as is known in the art.

Distal end 22 of catheter 20 further includes a device 28 that generates signals
used to determine the position and, preferably, orientation of the catheter within the
body. Device 28 is preferably adjacent to functional portion 24, in a fixed relation
with tip 26. Device 28 preferably comprises three non-concentric coils, such as
described in PCT patent publication W096/05768, whose disclosure is incorporated
herein by reference. This device enables continuous generation of six dimensions of
position and orientation information with respect to an externally-applied magnetic
field. Alternatively, device 28 comprises other position and/or coordinate sensors as
described in U.S. patent 5,391,199, U.S. patent 5,443,489 and PCT publication
WO94/04938, which are incorporated herein by reference. Further alternatively or
additionally, tip 26 is marked with a marker whose position can be determined from
outside of the body, for example, a radio-opaque marker for use with a fluoroscope.

Catheter 20 preferably includes a handle 30, having controls 32 which are used
by a surgeon to steer distal end 22 of the catheter in a desired direction, so as to
position and/or orient it as desired. Catheter 20 preferably comprises a steering
mechanism in distal end 22, as is known in the art, so that repositioning of tip 26 is
facilitated.

Catheter 20 is coupled, via an extension cable 21, to a console 34 which
enables the user to observe and regulate the functions of catheter 20. Console 34
preferably includes a computer 36, keyboard 38, signal processing circuits 40, which
are typically inside the computer, and display 42. Signal processing circuits 40
typically receive, amplify, filter and digitize signals from catheter 20, including
signals generated by position signal generating device 28, whereupon these digitized
signals are received and used by computer 36 to compute the position and orientation
of the catheter. Alternatively, appropriate circuitry may be associated with the catheter
itself so that circuits 40 receive signals that are already amplified, filtered and/or
digitized. Preferably, computer 36 includes a memory for storing positions and
determined parameters of the points. Computer 36 preferably also includes dedicated
graphic hardware for polygon manipulation, which allows performing reconstruction
stages described hereinbelow using fast computer graphic techniques.

Preferably, system 18 also includes an ECG monitor 73, coupled to receive
signals from one or more body surface electrodes 52 and to convey the signals to
computer 36. Alternatively, the ECG monitoring function may be performed by
circuits 40.

Fig. 2 shows a distal portion of mapping catheter 20 within a heart 70 of a
patient, in accordance with a preferred embodiment of the present invention. Catheter
20 is inserted into heart 70 and tip 26 is brought into contact with a plurality of
locations, such as locations 75 and 77 on an inner surface 72 of heart 70. Surface 72
bounds the volume to be reconstructed, and it is locations on this surface which are to
be sampled. At each of the plurality of locations, the coordinates of tip 26 are
determined by device 28, preferably together with physiological information
determined by functional portion 24. The determined coordinates and, optionally,
physiological information form a local data point. The local data points from a
plurality of locations are used for producing a map of heart 70, or of a portion of the
heart.

At least one reference catheter 78 is preferably inserted into heart 70 and is
placed in a fixed position relative to the heart. By comparing the positions of catheters
20 and 78, the position of tip 26 is accurately determined relative to the heart,
irrespective of heart motion. Alternatively, any other suitable method may be used to
compensate for movement of heart 70.

Preferably, the coordinates of tip 26 at the plurality of locations are determined
at a common time-point in the cardiac cycle, preferably at end-diastole. Alternatively
or additionally, each determined position is recorded together with a time-point,
preferably relative to a predetermined time-point in the cardiac cycle, and together
with indication of the current heart rate. The relative time-point and the rate of the
cycle are used to correct for the movement of the heart. Thus, it is possible to
determine positions of a large number of points, simply, in a limited time period.

Further alternatively or additionally, the position of tip 26 is determined at
each location at two or more time-points in the cardiac cycle, such that for each
location, a range of positions are determined. Thus, a geometric map of the plurality
of locations may comprise a plurality of "snapshots" of heart 70, each snapshot
associated with a different phase of the cardiac cycle. The cardiac cycle is preferably
determined using ECG monitor 73, according to physiological readings from
functional portion 24, or according to movements of reference catheter 78. Preferably,
each position is determined together with the heart rate at the time of determination. A
frequency and phase shift transformation is preferably applied to the plurality of
positions at each location to bring the positions to a state as if they were determined at
common time-points with respect to a common predetermined heart rate.

Preferably, the transformation applied to the positions also serves to reduce or
eliminate the effects of any movement of the heart that is not due to the cardiac cycle,
particularly chest movement due to respiration or other movements of the patient.
These effects are removed by defining a cyclic trajectory of the points associated with
each location, and then filtering out of the trajectory frequencies of motion other than
frequencies associated with the heart rate. Preferably, any frequencies whose
corresponding wavelengths do not evenly divide the cardiac cycle length, as
determined from the ECG, are filtered out. The result for each location is a modified
trajectory, including a corrected end-diastolic point, which is then used in
reconstructing the map of the heart, as described hereinbelow.

Preferably, at each location at which tip 26 is positioned, it is verified that
catheter 20 is in contact with the surface, using any suitable method, for example, as
described in PCT publication WO97/24981, which is incorporated herein by
reference.

Fig. 3 is a flow chart illustrating the process of point sampling and
reconstruction of a map, in accordance with a preferred embodiment of the present
invention. As described above, catheter 20 is brought into contact with surface 72 of
heart 70, and signals are received from the catheter to form a local data point
characteristic of the location of tip 26. The local data point preferably includes
coordinates of the point at a plurality of time points and one or more values,
associated with the point, of at least one physiological parameter. Preferably, as
mentioned above, the local data point includes an indication of the heart rate and time
point in the heart cycle for each determined coordinate. The parameter values may be
associated with specific time points or may be associated generally with the point.

Preferably, the contact between tip 26 and surface 72 is verified and the point
is added to the map only if there is sufficient contact between the tip and the surface.
In a preferred embodiment of the present invention, points for which proper contact
does not exist are added to a database of interior points. These points are interior to
the reconstructed surface and indicate areas on the map which are not part of the
reconstructed surface. Alternatively or additionally, the user may indicate sampled
points which are not to be used as part of the reconstructed surface, for example
because they are outstandingly outside of the area of the other sampled points. Tip 26
is then moved to an additional location on surface 72 and data are likewise determined
regarding the additional point. This procedure is repeated for a plurality of sampled
points until data are determined for a sufficient number of points to make the map, or
for a predetermined amount of time. Preferably, computer 36 counts the number of
sampled points and compares the number of points to a predetermined required
minimum number of points. Preferably, the predetermined number of points is
between about ten to twenty points for fast procedures and is up to 100 points for
longer procedures. Alternatively or additionally, the physician notifies computer 36
when a sufficient number of points have been sampled.

A map of heart 70 or of a volume within the heart is reconstructed, as
described below, and the physician decides whether the map includes sufficient detail
and appears to be accurate. If the map is not sufficient, more points are acquired and
the map is accordingly updated or is again reconstructed. The reconstructed map is
thereafter used for analysis of the functioning of heart 70, and the physician may
decide on a required treatment accordingly.

Fig. 4 is a flow chart illustrating a reconstruction procedure, in accordance
with a preferred embodiment of the present invention. Reconstruction is initially
performed for positions determined at an anchor time point (t0) of the heart cycle, such
as end diastole. In a first stage of the initial reconstruction, a grid enclosing the
sampled points is constructed. Thereafter, a stage of model distortion is applied to the
grid, in which the grid is roughly adjusted to the shape defined by the sampled points.
Subsequently, a preferably iterative stage of flexible matching is carried out finely
adjusting the grid points according to the coordinates of the sampled points. Final
adjustment is preferably applied to the grid including smoothing, an affine
transformation and/or an exact matching stage which brings the grid to include
substantially all the sampled points. The parameter values associated with the sampled
points are preferably interpolated to all the grid points and the grid is subsequently
displayed. This procedure is described in greater detail hereinbelow with reference to
the figures that follow.

Figs. 5A - 5E are simplified, two-dimensional graphs illustrating the
reconstruction procedure for a single time-point, in accordance with a preferred
embodiment of the present invention. For clarity of illustration, the figures and the
following description refer to a simplified, two dimensional example. The extension
of the principles illustrated herein to 3D reconstruction will be clear to those skilled in
the art. Points Si are sampled points on the surface of the volume to be reconstructed.
whose coordinates were received during the above-described sampling process.

As shown in Fig. 5A, in the first stage, an initial grid 90 is defined in a vicinity
of the sampled points, preferably enclosing the sampled points. Alternatively, grid 90
may be interior to the sampled points or pass between the points. Preferably, grid 90
comprises a number of points substantially greater than the number of sampled points.
The density of the points is preferably sufficient to produce a map of sufficient
accuracy for any required medical procedure. In a preferred embodiment of the
present invention, the physician can adjust the density of points on the grid according
to a desired compromise between reconstruction speed and accuracy. Preferably, grid
90 has an ellipsoidal shape or any other simple closed shape.

Alternatively or additionally, grid 90 has a shape based on known
characteristics of the volume on whose surface the sampled points are located, for
example, a shape determined by processing an LV-gram or other fluoroscopic or
ultrasound image of the heart. In a preferred embodiment of the present invention,
computer 36 contains a data-base of initial grids according to commonly-sampled
volumes. The physician indicates, preferably via keyboard 38, which volume is being
sampled and initial grid 90 is chosen accordingly. The chosen grid may be initially
aligned with the sample points using any method known in the art, for example as
described in Paul J. Besl and Neil D. McKay, "A method for registration of 3-D
shapes." IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(2):239-258,
February 1992, which is incorporated herein by reference. The initial grid may
alternatively be chosen from the grid library using geometric hashing or alignment, as
described, for example, in Haim J. Wolfson, "Model-based object recognition by
geometric hashing," in: O. Faugeras. ed., Computer Vision-ECCV90 (First European
Conference on Computer Vision, Antibes, France, April 23-27, 1990), Springer,
Berlin, 1990, 526-536, or in P. Huttenlocher and S. Ullman, "Recognizing solid
objects by alignment with an image," International Journal of Computer Vision, 5:
195-212, 1990, which are incorporated herein by reference. After the initial
alignment, the method of the present invention proceeds, preferably as shown in Fig. 4
and described further hereinbelow.

As shown in Fig. 5B, grid 90 is transformed to a grid 92 of points G', which is
a rough adjustment toward the structure of the sampled volume. For each point Gj on
grid 90, an adjustment vector Vj is constructed, and point Gj is replaced by a
corresponding point Gj' on grid 92, which is displaced by Vj from point Gj on grid
90. Adjustment vector Vj is preferably a weighted sum of vectors Vji from Gj to the
sampled points Si, as shown in Fig. 5A. Preferably, the weights of vectors Vji in the
sum are strongly inversely dependent on the magnitude of the vectors. Preferably, the
weights are inversely dependent on the magnitude raised to a power (k), wherein k
preferably ranges between 4 and 10, and is most preferably either between 6 and 8. In
a preferred embodiment of the present invention, adjustment vectors Vj are
calculated according to equation (1):

In equation (1), epsilon is a small scalar, preferably, smaller than the
magnitude of the smallest vector which is not zero, and is preferably of the size of the
accuracy of the determination of the sampled points, for example, about 10-6. Epsilon
is used to prevent division by zero when the grid point is on a sampled point, and
therefore the magnitude of the vector is zero. Cf is a constant factor between 0.1 and
1, preferably between 0.5 and 0.9 most preferably about 0.75, which is adjusted to
determine how closely the points Gj' will approach points Si in the rough adjustment.

In a preferred embodiment, the influence of a sampled point Si on grid point
Gj, takes into account not only the distance between the sampled point Si and Gj, as
shown above in equation (1) but also the density of sampled points S in the vicinity of
Si. Hence, the weighting factor applied to each sampled point,
1rkj + ε ,
is multiplied
by a density value δi, which preferably takes on values between 0 and 1. Preferably, δi
is as defined in equation (2):

The more points there are in the vicinity of S, the smaller value δ takes on and
the less influence each point has. Preferably, the sum of influences of a plurality of
points in a close vicinity is the same as the influence of a single isolated point, which
preferably has a density value δ of about 1.

Fig. 5C illustrates a first pan of a flexible matching step, in which each of
sampled points Si is associated with a grid point Gj from roughly adjusted grid 92.
The associated grid points are moved toward their respective sampled points, while
the rest of the G' points on the roughly adjusted grid are moved according to
interpolation of the movements of neighboring points on grid 92, as described further
hereinbelow. Preferably, each sampled point Si is associated with the closest grid
point. For example, the closest grid point to S1 is G1', and these points are therefore
associated. Preferably, computer 36 creates a memory list in which these pairs of
points are listed. For clarity of this explanation, the associated points are marked by
dashed ovals 96 in Fig. 5C.

Preferably, a transformation function f, which moves the associated grid points
toward their respective sampled points, is generated. The non-associated grid points
are also moved according to function f. Function f is preferably easily calculated, and
transforms the grid to a smooth form. Preferably, function f is a weighted sum of the
distances between the associated pairs of sampled and grid points, such that pairs of
associated points close to the grid point influence its displacement more than pairs of
associated points far from the grid point. Function f is preferably as given in equation
(3) below, with wi(Gj) dependent on the distances between the grid point Gj and the
associated grid points Gi, preferably as defined in equation (4). Alternatively, wi(Gj)
is dependent on the distance between the grid point Gj and the sampled points Si, as in
equation (1). In the flexible matching stage, k is preferably smaller than the power law
in the rough adjustment stage in order to generate a smoother grid surface. Preferably,
k in the flexible matching stage is between 2 and 6 and is most preferably 4.
Preferably, k is an even number in order to simplify the calculations. Although the
equations below are stated for convenience in scalar notation, it will be understood
that Si, Gi and f(Gj) are vector quantities, as in equation (1) above:

wi(Gj) = 1Gj - Sik + C C>0

The constant C determines how close the associated grid points are moved
toward their associated sampled points. For very small values of C, the associated grid
points Gi are moved substantially onto the sampled points Si. Preferably, C is between
0.3 and 0.7, more preferably about 0.5. Alternatively or additionally, C is changed
according to the number of times the flexible matching is to be performed. Further
alternatively or additionally, in the first flexible matching step, C is relatively large,
while in subsequent flexible matching steps C is gradually reduced.

The distance definition used in equations (2), (3) and (4) is preferably the
Euclidean distance in R3, due to its simplicity in calculation and the fact that it causes
points on opposite walls of the reconstructed volume to repel one another.

In an alternative preferred embodiment of the present invention, the grid points
which have an associated sampled point are moved toward their associated sampled
points by a portion of the distance between them. Preferably, the points are moved a
percentage of the distance between the associated pair. For example, in Fig. 5C the
points are moved about 2/3 of the distance. Alternatively, the grid points are moved
by any other amount dependent on the distance between the associated pair.

As shown in Fig. 5D, those grid points G'k which are not associated with
sampled points Si are then moved according to a movement vector Vk which is
dependent on the movements of the grid points G'1 surrounding the point. Preferably,
the non-associated points G'k are moved a distance which is a linear interpolation of
the movements of the surrounding points G'1. Preferably, the distance between the
grid points is determined as the geometrical distance between the points as they are on
the present adjusted grid. For example, the geometrical distance between G'15 and
G'16 is indicated by X2, and may be calculated according to the coordinates of the
two points. Alternatively or additionally, the distance used is the grid-distance

along the present adjusted grid, the grid-distance

along the original
grid, or the geometrical distance L2 on the original grid. In a preferred embodiment of
the present invention, in a first flexible matching step, the distance used is the grid-distance
- either l2 or - while in subsequent flexible matching steps the distance
used is the geometrical distance X2.

For example, as shown in Fig. 5D, point G'15 is moved a distance defined by a
vector , which is a weighted sum of vectors V14, andV16 of grid points G'14, and G'16,
respectively. Preferably, V15 is as described in equation (2) below, in which d1 is a
selected type of distance between G15 and G14, and may include X1,

, l1 or any
other suitable distance definition. Likewise, d2 is a selected type of distance between
G15 and G16 and may include X2, , l2, or any other distance definition.
Preferably, in the first flexible matching step illustrated in Fig. 5D, d1 and d2 are
taken as X1 and X2 respectively.
V'15 = d2d1 +d2V'14 + d1d1 + d2V'16

Although equation (8) illustrates a first-order linear interpolation, it will be
understood that higher-order and non-linear interpolation methods may also be used.

Preferably, during the flexible matching stage. flexible matching steps are
repeated a few times (N0 times, as shown in Fig. 4). Each time, grid points are
associated with the sampled points, and the associated and non-associated grid points
are moved accordingly.

The rough adjustment and flexible matching tend to cause the grid to become
non-uniform. Therefore, during a final adjustment stage the grid is preferably
smoothed, for example, by applying a surface convolution with a Gaussian-like
kernel. Preferably, the kernel is a 3x3 Gaussian kernel, and is applied to the grid a
plurality of times, preferably between five and ten times. Alternatively, a larger kernel
may be used in which case it may be applied to the grid fewer times, most preferably
only once. The surface convolution, however, generally causes shrinkage of the
surface, and therefore a simple transformation, preferably an affine transformation, is
applied to the grid to cancel the shrinkage and improve the matching of the grid to the
sampled points. The affine transformation is preferably chosen as the transformation
which minimizes the mean square distance between sampled points outside of the grid
and a surface defined by the grid. This choice of the transformation causes
substantially all the sampled points to be on or inside the surface defined by the grid.
This choice is in accordance with the anatomical structure of the heart in which
outliers, i.e., points not on the sampled surface, are generally inside the sampled
surface, i.e. inside a cardiac chamber rather than on the myocardial wall. Thus, the
reconstructed grid is properly reconstructed by ignoring outliers which otherwise may
deform the grid incorrectly.

To conclude the final adjustment stage, the user may optionally request an
exact matching stage in which the grid surface is deformed to include substantially all
the sampled points. Preferably, for each sampled point not on the grid surface as a
result of prior stages, a closest grid point is chosen and moved to the position of the
sampled point. The rest of the grid points are preferably not moved. Preferably,
internal points which are beyond a certain distance from the grid surface are not
moved in this stage and are regarded as outliers. It is noted that external points are not
generally distanced from the grid surface due to the affine transformation described
above.

Alternatively or additionally, a last flexible matching step is performed in
which the associated grid points are moved onto the sampled points, as shown in Fig.
5E. Curved line 100 in Fig. 5E represents the final grid configuration and comprises
an accurate approximation of the sampled volume.

Alternatively the flexible matching is performed in one step, and the
associated points from the rough adjustment grid are immediately moved onto the
sampled points. In a preferred embodiment of the present invention, computer 36 first
produces an approximate map, in which the flexible matching is performed in one
step. The approximate map is used by the physician to decide if more sampled points
are needed. Once the physician decides that no more points are needed, computer 36
reconstructs a more accurate map in which the flexible matching is performed a
plurality of times. Meanwhile, the physician may use the approximate map in order to
save time. In further preferred embodiments, the first reconstructed map is produced
with a relatively low density of points on the grid, while later reconstructions use a
more dense grid.

Referring back to Fig. 4, when the sampled points include data from more than
one time point, the reconstructed grid of the anchor time point (hereinafter referred to
as the anchor grid) is preferably used to quickly reconstruct the grid for other time
points ti. For each of the other time points, a simple transformation is performed on
the anchor grid to bring the grid close to the form of the sampled points of time ti. The
simple transformation is preferably a quadratic transformation or an affine
transformation. Alternatively, the transformation comprises a rotation and/or scaling
transformation. In some preferred embodiments of the present invention, the
transformation is chosen according to the number of sampled points. Preferably, when
there are a relatively large number of sampled points, a quadratic transformation is
applied, while for fewer sampled points, simpler transformations are employed.

Flexible matching is then preferably performed on the transformed grid one or
more times (NT), preferably fewer times than were required in reconstruction of the
anchor-time grid (NT<N0), most preferably twice. Final adjustments are then
preferably applied to the grid, and the resulting grid at time t1 may be displayed. The
parameter value may also be interpolated separately for time t1, substantially as
described above with respect to the anchor grid. When reconstruction for all of the
time points is concluded, the reconstructed grids may be displayed in sequence as a
function of time, or in any other manner. Preferably, the reconstruction process
continues while the anchor grid is displayed, so that a physician may use the
reconstructed data without delay.

Preferably, as noted hereinabove, each data point includes at least one
physiological parameter, such as an indicator of the electrical activity in the heart,
measured using functional portion 24 of catheter 20. After the map is constructed, as
described above, the points on the grid, G1, G'4, G'7, etc., that were associated with
sampled points S1, S2, S6, etc., are assigned the physiological parameter value of
their respective sampled points. The non-associated grid points receive parameter
values by interpolation between the values of the parameters of neighboring
associated grid points in a manner similar to that described above. Alternatively or
additionally, the non-associated grid points receive parameter values in a manner
similar to the way they received their coordinates in flexible matching.

Further alternatively or additionally, the non-associated grid points are given
parameter values using a zero-order-hold filling in method. Starting from the sampled
points, all the surrounding grid points are given the same parameter value as the
sampled point has, propagating outward until another grid point with a different
parameter value is encountered. Thereafter, a Gaussian smoothing process is
preferably applied to the parameter values. Thus, parameter values are given in a very
simple method to all the grid points substantially without forfeiting visual clarity.

Thus, a 3D map is reconstructed showing both the geometrical shape of the
heart chamber and local electrical parameters or other physiological parameters as a
function of position in the heart. The local parameters may include electrogram
amplitude, activation time, direction and/or amplitude of the electrical conduction
vector, or other parameters, and may be displayed using pseudocolor or other means
of graphic realization, as is known in the art. Preferably, a predefined color scale is
associated with the parameter, setting a first color, e.g., blue, for high values of the
parameter, and a second color, e.g., red. for low values of the parameter.

Fig. 6 is a schematic illustration of a displayed reconstructed heart volume
130, in accordance with a preferred embodiment of the present invention. A plurality
of sampled points 134 are used to reconstruct a surface 132 of volume 130. A grid
(not shown) is adjusted as described above to form surface 132. Preferably, each point
on the grid receives a reliability value indicative of the accuracy of the determination.
Further preferably, the reliability value is a function of the distance from the grid
point to the closest sampled point on surface 132 and/or of a density of sampled points
134 in a vicinity of the grid point. Preferably, areas of surface 132 covered by less-reliable
grid points, such as an area 140, are displayed as semi-transparent, preferably
using α-blending. Due to the transparency, points 136 on an inner surface of volume
130 are displayed, being seen through volume 130. Preferably, the user may define the
predetermined distance and/or sample density defining less-reliable points.
Alternatively or additionally, different levels of semi-transparency are used together
with a multi-level reliability scale.

Fig. 7 is a schematic illustration of a volume estimation method, in accordance
with a preferred embodiment of the present invention. In some cases it is desired to
estimate the volume encompassed by one or more reconstructed surfaces, for example,
to compare the volume of a heart chamber at different time-points of the heart cycle.
In Fig. 7 the reconstructed grid surface is represented, for clarity, by a ball 150. The
surface of ball 150 is partitioned into quadrilaterals by the grid points, and these
quadrilaterals are used for volume estimation. An arbitrary point O, in a vicinity of the
surface, preferably within the volume, most preferably close to the center of mass of
ball 150, is chosen, thus defining a pyramid 152 for each quadrilateral on the surface
of ball 150. An estimate of the sum of the volumes of pyramids 152 accurately
represents the volume of ball 150.

Preferably, each quadrilateral is divided into two triangles, and the volume is
estimated by summing the volumes of tetrahedrons defined by these triangles as bases
and vertex O apex. Let Am, Bm, Cm, denote the vertices of the m-th triangle arranged
clockwise, so that the normals of the triangles point outward from the surface of ball
150. The volume V of ball 150 is estimated by equation (6):

Fig. 8 is an illustration of a reconstruction procedure, in accordance with
another preferred embodiment of the present invention. In this preferred embodiment
the sampled points are known to be on a single, open surface, rather surrounding a 3D
volume, and therefore the beginning grid may comprise an open plane, rather than a
closed curve. Catheter 20 is brought into contact with a plurality of locations on an
inner wall 76 of heart 70, and the coordinates of these locations are determined to give
sampled points 120. Preferably, a physician indicates to console 34 the direction from
which catheter 20 contacts surface 76. Computer 36 accordingly generates an initial
grid 122, which includes a plurality of grid points 124, such that all the grid points are
preferably on one side of the sampled points. The adjustment procedure is performed
substantially as described above, bringing grid points 124 to maximally resemble
surface 76.

In a preferred embodiment of the present invention, the adjustment procedure
may be performed step-by-step on display 42, allowing the physician to interrupt and
direct the procedure if necessary.

It is noted that although the above description assumes that the data regarding
the sampled points are acquired by the system which performs the reconstruction, the
reconstruction procedure may also be performed on points received from any source,
such as from a different computer, a library database or an imaging system.
Furthermore, although preferred embodiments are described herein with reference to
mapping of the heart, it will be appreciated that the principles and methods of the
present invention may similarly be applied to 3D reconstruction of other physiological
structure and cavities, as well as in non-medical areas of 3D image reconstruction.

As noted above, an important example of a physiological parameter of the
heart, that is measured using functional portion 24 of catheter 20 and that is assigned
to the grid points that are associated with the sampled points, is the local activation
time (LAT) of the heart tissue. This time is determined by referring the time of a
feature of the signal (specifically, a voltage) measured by functional portion 24 at
each sampled point, for example, the time in the cardiac cycle at which that signal
first exceeds a certain threshold, to the time within the cardiac cycle of a fiducial
feature of the ECG signal, as measured, for example, using ECG monitor 73.
Preferably, the grid on which LAT is posted is the grid corresponding to end diastole,
because the heart is most fully expanded at that point in the cardiac cycle, and the
interior surfaces of the chamber of the heart consequently are smoothest at that point
in the cardiac cycle.

The values of LAT, that are posted at the grid points associated with the
sampled points, are interpolated to the other grid points, as described above.
Preferably, this interpolation is done using a variant of the zero-order-hold filling
method, based on the distance d(V) from each grid point V to the nearest sampled
points, as measured along the grid.

Initially, the grid points that coincide with sample points are assigned d(V)
values of zero, and all the other grid points are assigned d(V) values of infinity. Then,
in each of a sequence of iterations, each grid point V is visited in turn, and is assigned
a new value of d(V), based on the distance d(V,Ni) between that grid point V and its m
neighboring grid points Ni ∈ {N1,...,Nm}. Specifically, d(V) is replaced with

As each grid point V is assigned a new value of
d(V), that grid point V also is assigned the LAT value associated with the neighbor Ni
upon which the new value of d(V) is based. These iterations are continued as long as
at least one d(V) changes in the course of an iteration. Finally, the posted LAT values
are smoothed by convolution, as described above in the context of the final adjustment
of grid geometry.

The preferred 3D grid is one in which the grid points are connected by lines in
a way that defines the grid as a collection of polygons, for instance triangles, with the
grid points constituting the vertices of the triangles and with the lines connecting the
grid points constituting the edges of the triangles. In such a grid, a preliminary
version of the propagation velocity of the activation signal, i.e., the conduction
velocity of the heart tissue, is obtained by assigning a velocity vector to each triangle,
based on the LAT values at the triangle's vertices. It is assumed that the grid is
sufficiently fine that, in each triangle, the activation signal propagates as a plane
wave. Fig. 9 shows a triangle 200 with vertices a, b and c , and with a planar
wavefront 202 propagating across triangle 200 towards the upper right at a velocity v.
Note that wavefront 202 is perpendicular to the direction of propagation. Wavefront
202 is shown at the time tb at which wavefront 202 reaches vertex b. This time is at
least as great as the time ta at which wavefront 202 reached vertex a, and is no greater
than the time tr at which wavefront 202 will reach vertex c : ta≤tb≤tc. Wavefront 202
intersects side ac of triangle 200 that is opposite vertex b at a point d. Point d is
found by linear interpolation:
d = tb -tdtc -tac + tc -tbtc -taa
The unit vector in the direction of v is found by taking the cross product of d-b
with the unit vector N normal to triangle 200 and normalizing:
vv= d -bd-b × N
Finally, the magnitude of v is found by projecting the apparent velocity from a to c
onto this unit vector:
v= c-atc -ta • vv

Having thus assigned a velocity vector to each triangle of the grid, each grid
point is assigned a raw velocity vector by averaging the velocities of all the triangles
of which that grid point is a vertex. Finally, the raw velocities are smoothed
iteratively, as follows:

1. Each triangle is assigned, as a new velocity, the average of the
velocities assigned to the grid points which are the vertices of the triangle.

2. Each grid point is assigned, as a new velocity, the average of the
velocities assigned to the triangles of which the grid point is a vertex.

Preferably, the conduction velocity vector function thus obtained is displayed
superposed on a display of the surface represented by the grid, both as a pseudocolor
map, as described above, or as arrows emerging from the grid points. In one variant
of this display, the direction of the arrow at each gnd point corresponds to the
direction of v as posted and smoothed at that grid point; and the length of the arrow
corresponds to the magnitude of v as posted and smoothed at that grid point.
Alternatively, all the arrows have the same length, and the arrows are displayed in
monochrome or achromatic manner, using a gray scale that encodes the magnitudes of
Alternatively, the arrows may be displayed according to a specific color scheme.
The iterative smoothing parameters may be determined by a priori knowledge of the
specific heart.

It will be appreciated that any vector function that is derived from a set of
scalar measurements on the surface of a biological structure may be displayed in this
manner. Furthermore, the vector function may be displayed along with the scalar
measurements from which it was derived, or along with a scalar function of the scalar
measurements from which the vector function was derived. For example, LAT may
be displayed as a pseudocolor map, and the corresponding conduction velocity vector
function may be displayed as arrows superposed on the pseudocolor map, as described
above.

Fig. 10 shows such a display of a normal human atrium. LAT is normally
displayed as a scale in pseudocolor, but is herein depicted with a numerical scale. The
numerical scale with respect to the LAT ranges from a minimum (1) which is the
earliest activation time, to a maximum (10) which is the latest activation time. The
direction of the corresponding conduction velocity vector field is shown by the
arrows. The arrows are displayed in monochrome, with the gray scale level of each
arrow corresponding to the magnitude of the associated conduction velocity vector.
As is shown in the lower left hand portion of the figure, the velocity magnitude scale
ranges from a minimum (solid black arrow) to a maximum (open headed arrow).
Midrange is shown with a dotted arrow. The signal flow is predominantly radially
away from the region numbered one (1) in which activation is initiated.

Fig. 11 is a similar display of LAT and conduction velocity in a human atrium
suffering from atrial flutter. The signal flow tends to be vortical, rather than radially
outward. This vortical flow is evidenced by the distinct and separate patterns of
conduction velocity vector arrows shown.

Fig. 12 shows a pattern on such a display that is diagnostic of ventricular
tachycardia: a region of scar tissue associated with a vortical conduction velocity field
that is represented by circular patterns of arrows. LAT is shown with a numerical
scale from 1 to 10. A physician treats ventricular tachycardia thus diagnosed by
ablating the heart tissue in the region of the pattern shown in Fig. 12. Such a display
also provides quality control diagnostics, inasmuch as the magnitude of the
conduction velocity is expected to be abnormally low in scar tissue.

Fig: 13 shows the conduction velocity vectors alone (without display of LAT
regions) in the left ventricle of a dog. The heart is entrained in a sinus rhythm from
the right ventricle apex. The velocity vector arrows are distributed according to the
density of the underlined grid. Each arrow represents the local conduction velocity.
The arrow direction is the computed direction of the conduction and its gray scale
color represents the conduction velocity magnitude (black colored arrows indicate
slow conduction velocity, gray colored arrows indicate midrange conduction velocity
and white colored arrows indicate fast conduction velocity).

Fig. 14 is the right atrium of a human heart suffering from atrial flutter. The
conduction velocity vectors are also depicted alone, e.g. without display of LAT
regions or other parameters. Rather than having a well-defined focus that starts the
activation in the heart, such as that found in the heart example depicted in Fig. 10, the
cardiac wave, as depicted by the conduction velocity vectors, moves in distinct
circular patterns. These circulated patterns result in a convergence of the cardiac
wave as shown along the lower central portion of the atrium. One type of treatment
involves ablations along this area of the atrium in order to disable the abnormal
circuitry. After ablation, the chamber can be remapped to ensure that the procedure
has been performed successfully.

Other scalar functions of the ECG measurements used to derive LAT also are
useful. One such scalar function is the amplitude (maximum - minimum) of voltages
measured at each sampled point over the course of the cardiac cycle. A low amplitude
is diagnostic of scar tissue. Most preferably, voltage amplitude, LAT and conduction
velocity are displayed together. Voltage amplitude is encoded in a conventional
pseudocolor map. LAT is encoded as colored dots posted on the sampled points.
Conduction velocity is displayed as arrows, as described above.

As mentioned previously, once the conduction velocity vectors, (indicated by
arrows), are displayed superimposed on the 3D map of the surface of the heart,
treatment may be administered to those areas of the heart depicted as being
problematic based on the displayed velocity vectors. For instance, ablative treatment
is administered at those areas depicting velocity vector direction, e.g. converging
arrows such as shown in Figs. 11 and 14. It is within the scope of the present
invention to include any type of treatment modality such as the application of energy,
for example laser, therapeutic ultrasound, radiofrequency, etc. as well as
pharmaceutical or biological therapy. Moreover, therapeutic treatment may be
administered based on the magnitude of the velocity vectors. For instance, in the gray
scale embodiment, those velocity vector arrows that are identified by the color black
indicate low conduction velocity. Since the propagation wave is identified to move
slowly through this portion of the heart, this may be indicative of diseased tissue or
scar tissue.

Another useful quality control diagnostic is obtained by displaying yet a third
scalar field. This scalar field is obtained by performing calculations of conduction
velocity as described above, but excluding, from each calculation, one of the sampled
points, with a different sampled point being excluded from each calculation. This is
done for each sampled point, thereby producing as many calculations of the
conduction velocity field as there are sampled points. The associated scalar field is, at
each grid point, the range (maximum - minimum) of conduction velocity magnitudes
obtained at that grid point. This scalar field, displayed in pseudocolor, provides a
measure of the reliability of the calculated conduction velocity field at each grid point.

It is also possible to display the conduction velocity with other physiological
maps, for example, the voltage map or the impedance map, generated for the same
recordings of the organ.

It is noted that the above displays may be displayed in at least two ways: by a
color from the pseudocolor scale when the value represents one that is of a determined
confidence level and as such, may be placed directly on the pseudocolor map; and by
another, different color or transparency, when the value is of low confidence and as
such, is so displayed on the map. In the latter case, the practitioner will be guided to
acquire more samples.

It will thus be appreciated that the preferred embodiments of the invention
described above are cited by way of example, and the full scope of the invention is
limited only by the claims which follow.

Claims (16)

A method of displaying a condition in a heart, comprising the steps of:

(a) creating a map of a surface of the heart;

(b) measuring a physiological response at at least three sampled points on
the surface of the heart;

(c) calculating a vector function related to said response; and

(d) displaying a representation of said vector function on said map.

The method of claim 1, wherein said vector function is related to a
gradient of said physiological response.

The method of claim 2, wherein said physiological response is a
function of time.

The method of claim 3, wherein said physiological response is a local
activation time (LAT) of a physiological signal propagating in the heart.

The method of claim 4, wherein said vector function is a velocity of
said propagation.

The method of claim 1, wherein said representation of said vector
function includes an arrow at each of said sampled points.

The method of claim 6, including representing said surface of said
heart as a grid including a number of grid points, said number of grid points being at
least as great as said at least three sampled points.

The method of claim 7, wherein each grid point substantially coincides
with a corresponding one of said at least three sampled points.

The method of claim 8, wherein said grid includes a plurality of
polygons defined by a plurality of vertices, each vertex of each said polygon being
one of said grid points, and wherein said calculating of said vector function includes
the steps of:

(i) interpolating a local activation time (LAT) value at each grid point;

(ii) for each of said polygons, assigning a value of said vector function to
said each polygon, based on said interpolated LAT value at said grid
points that are vertices of said each polygon; and

(iii) for each of said grid points, determining a value of said vector function
for said each grid point, based on said values of said vector function
that are assigned to said at least one polygon whereof said each grid
point is a vertex.

The method of claim 9, wherein said calculating of said vector function
further includes the step of:
(iv) smoothing said values of said vector function at said grid points.

The method of claim 10, wherein an extent of said smoothing is based
on a priori information about the condition.

The method of claim 11, wherein said polygons are triangles.

The method of claim 6, wherein said arrow indicates a direction of said
vector function at each sampled point.

The method of claim 13, wherein said arrow has a length which
indicates a magnitude of said vector function at each sampled point.

The method of claim 13, wherein said arrow is depicted according to a
gray scale which indicates a magnitude of said vector function at each sampled point.

The method of claim 13, wherein said arrow is depicted according to a
color scheme which indicates a magnitude of said vector function at each sampled
point.

The catheter system for analyzing data system, measured from the working methods and heart tissue of a system for analyzing the electrophysiological data for mapping analyzes electrophysiological data from the body tissue